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A decomposition of a set definable in an o-minimal structure into perfectly situated sets

Wiesław Pawłucki (2002)

Annales Polonici Mathematici

A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such that any...

A first-order version of Pfaffian closure

Sergio Fratarcangeli (2008)

Fundamenta Mathematicae

The purpose of this paper is to extend a theorem of Speissegger [J. Reine Angew. Math. 508 (1999)], which states that the Pfaffian closure of an o-minimal expansion of the real field is o-minimal. Specifically, we display a collection of properties possessed by the real numbers that suffices for a version of the proof of this theorem to go through. The degree of flexibility revealed in this study permits the use of certain model-theoretic arguments for the first time, e.g. the compactness theorem....

A fixed point theorem in o-minimal structures

Mário J. Edmundo (2007)

Annales de l’institut Fourier

Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps.

A gradient inequality at infinity for tame functions.

Didier D'Acunto, Vincent Grandjean (2005)

Revista Matemática Complutense

Let f be a C1 function defined over Rn and definable in a given o-minimal structure M expanding the real field. We prove here a gradient-like inequality at infinity in a neighborhood of an asymptotic critical value c. When f is C2 we use this inequality to discuss the trivialization by the gradient flow of f in a neighborhood of a regular asymptotic critical level.

A linear extension operator for Whitney fields on closed o-minimal sets

Wiesław Pawłucki (2008)

Annales de l’institut Fourier

A continuous linear extension operator, different from Whitney’s, for 𝒞 p -Whitney fields (p finite) on a closed o-minimal subset of ℝ n is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.

A Note on a Theorem of Lion

Zofia Ambroży (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

In this note we bind together Wilkie's complement theorem with Lion's theorem on geometric, regular and 0-regular families of functions.

A note on generic subsets of definable groups

Mário J. Edmundo, G. Terzo (2011)

Fundamenta Mathematicae

We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.

A proof of the valuation property and preparation theorem

Krzysztof Jan Nowak (2007)

Annales Polonici Mathematici

The purpose of this article is to present a short model-theoretic proof of the valuation property for a polynomially bounded o-minimal theory T. The valuation property was conjectured by van den Dries, and proved for the polynomially bounded case by van den Dries-Speissegger and for the power bounded case by Tyne. Our proof uses the transfer principle for the theory T c o n v (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den...

A theorem on generic intersections in an o-minimal structure

Krzysztof Jan Nowak (2014)

Fundamenta Mathematicae

Consider a transitive definable action of a Lie group G on a definable manifold M. Given two (locally) definable subsets A and B of M, we prove that the dimension of the intersection σ(A) ∩ B is not greater than the expected one for a generic σ ∈ G.

Algebraic approximation of analytic sets definable in an o-minimal structure

Marcin Bilski, Kamil Rusek (2010)

Annales Polonici Mathematici

Let K,R be an algebraically closed field (of characteristic zero) and a real closed field respectively with K=R(√(-1)). We show that every K-analytic set definable in an o-minimal expansion of R can be locally approximated by a sequence of K-Nash sets.

An o-minimal structure which does not admit C ∞ cellular decomposition

Olivier Le Gal, Jean-Philippe Rolin (2009)

Annales de l’institut Fourier

We present an example of an o-minimal structure which does not admit C ∞ cellular decomposition. To this end, we construct a function H whose germ at the origin admits a C k representative for each integer k , but no C ∞ representative. A number theoretic condition on the coefficients of the Taylor series of H then insures the quasianalyticity of some differential algebras 𝒜 n ( H ) induced by H . The o-minimality of the structure generated by H is deduced from this quasianalyticity property.

An ordered structure of rank two related to Dulac's Problem

A. Dolich, P. Speissegger (2008)

Fundamenta Mathematicae

For a vector field ξ on ℝ² we construct, under certain assumptions on ξ, an ordered model-theoretic structure associated to the flow of ξ. We do this in such a way that the set of all limit cycles of ξ is represented by a definable set. This allows us to give two restatements of Dulac’s Problem for ξ - that is, the question whether ξ has finitely many limit cycles-in model-theoretic terms, one involving the recently developed notion of U þ -rank and the other involving the notion of o-minimality.

Arithmetization of the field of reals with exponentiation extended abstract

Sedki Boughattas, Jean-Pierre Ressayre (2008)

RAIRO - Theoretical Informatics and Applications


 (1) Shepherdson proved that a discrete unitary commutative semi-ring A+ satisfies IE0 (induction scheme restricted to quantifier free formulas) iff A is integral part of a real closed field; and Berarducci asked about extensions of this criterion when exponentiation is added to the language of rings. Let T range over axiom systems for ordered fields with exponentiation; for three values of T we provide a theory ⌞ T ⌟ in the language of rings plus exponentiation such that the ...

Corps C-minimaux, en l’honneur de François Lucas

Françoise Delon (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

La classe des constructibles de la géométrie algébrique est close par projection. La théorie des modèles exprime ce fait en disant que les corps algébriquement clos éliminent les quantificateurs dans le langage des anneaux. De façon analogue, les corps algébriquement clos non trivialement valués éliminent les quantificateurs dans le langage des anneaux enrichi de la relation dite de divisibilité v ( x ) ≤ v ( y ) . Cela implique en particulier la «  C -minimalité » : une partie définissable d’un corps algébriquement...

Decomposition into special cubes and its applications to quasi-subanalytic geometry

Krzysztof Jan Nowak (2009)

Annales Polonici Mathematici

The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.

Definably complete Baire structures

Antongiulio Fornasiero, Tamara Servi (2010)

Fundamenta Mathematicae

We consider definably complete Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain cannot be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire, and so is every o-minimal expansion of a field. Moreover, unlike the o-minimal case, the structures considered form an axiomatizable class. In this context we prove a version of the Kuratowski-Ulam...

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